Problem 3.14. Experimental measurements of the heat capacity of aluminum at low temperatures (below about 50 K) can be fit to the formula Cy=aT+bT³, where Cy is the heat capacity of one mole of aluminum, and the constants a and b are approximately a = 0.00135 J/K² and b = 2.48 x 10-5 J/K4. From this data, find a formula for the entropy of a mole of aluminum as a function of temperature. Evaluate your formula at T = 1 K and at T = 10 K, expressing your answers both in conventional units (J/K) and as unitless numbers (dividing by Boltzmann's constant). [Comment: In Chapter 7 I'll explain why the heat capacity of a metal has this form. The linear term comes from energy stored in the conduction electrons, while the cubic term comes from lattice vibrations of the crystal.]
Problem 3.14. Experimental measurements of the heat capacity of aluminum at low temperatures (below about 50 K) can be fit to the formula Cy=aT+bT³, where Cy is the heat capacity of one mole of aluminum, and the constants a and b are approximately a = 0.00135 J/K² and b = 2.48 x 10-5 J/K4. From this data, find a formula for the entropy of a mole of aluminum as a function of temperature. Evaluate your formula at T = 1 K and at T = 10 K, expressing your answers both in conventional units (J/K) and as unitless numbers (dividing by Boltzmann's constant). [Comment: In Chapter 7 I'll explain why the heat capacity of a metal has this form. The linear term comes from energy stored in the conduction electrons, while the cubic term comes from lattice vibrations of the crystal.]
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![Problem 3.14. Experimental measurements of the heat capacity of aluminum at
low temperatures (below about 50 K) can be fit to the formula
Cv=aT+bT³,
where Cy is the heat capacity of one mole of aluminum, and the constants a
and b are approximately a = 0.00135 J/K² and b = 2.48 × 10-5 J/K4. From
this data, find a formula for the entropy of a mole of aluminum as a function of
temperature. Evaluate your formula at T = 1 K and at T = 10 K, expressing
your answers both in conventional units (J/K) and as unitless numbers (dividing
by Boltzmann's constant). [Comment: In Chapter 7 I'll explain why the heat
capacity of a metal has this form. The linear term comes from energy stored in
the conduction electrons, while the cubic term comes from lattice vibrations of the
crystal.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff69ee1b-fb03-4465-b15c-ee94e0c7648c%2F0aa1ff17-98e6-4ca3-a2a3-f23b078433f4%2Fb219g54_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 3.14. Experimental measurements of the heat capacity of aluminum at
low temperatures (below about 50 K) can be fit to the formula
Cv=aT+bT³,
where Cy is the heat capacity of one mole of aluminum, and the constants a
and b are approximately a = 0.00135 J/K² and b = 2.48 × 10-5 J/K4. From
this data, find a formula for the entropy of a mole of aluminum as a function of
temperature. Evaluate your formula at T = 1 K and at T = 10 K, expressing
your answers both in conventional units (J/K) and as unitless numbers (dividing
by Boltzmann's constant). [Comment: In Chapter 7 I'll explain why the heat
capacity of a metal has this form. The linear term comes from energy stored in
the conduction electrons, while the cubic term comes from lattice vibrations of the
crystal.]
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